- Bermudan options
- Asian options
- Lookback options
- Monte Carlo Pricing of options whose payoff depend on the whole price path

- Control variate

We now look at a type of options that has received a lot of attention in later years. The distinguishing factor of these options is that they depend on the whole price path of the underlying security between today and the option maturity.

A Bermudan option is, as the name implies,^{13.1} a mix of an European and American option. It is a standard put or call option which can
only be exercised at discrete dates throughout the life of the option. The simplest way to do the pricing
of this is again the binomial approximation, but now, instead of checking at every node whether it is
optimal to exercise early, only check at the nodes corresponding to the potential exercise times.
Code 13.1 shows the calculation of the Bermudan price using binomial
approximations. The times as which exercise can happen is passed as a vector argument to the routine, and
in the binomial a list of which nodes exercise can happen is calculated and checked at every step.

The payoff depends on the average of the underlying price. An average price call has payoff

where is the average of the underlying in the period between and .

Another Asian is the average strike call

There are different types of Asians depending on how the average is calculated. For the case of being lognormal and the average being a geometric average, there is an analytic formula due to Kemna and Vorst (1990). Hull (2003) also discusses this case. It turns out that one can calculate this option using the regular Black Scholes formula adjusting the volatility to and the dividend yield to

in the case of continous sampling of the underlying price distribution.

Code 13.2 shows the calculation of the analytical price of an Asian geometric average price call.

Lookback options

The payoff from lookback options depend on the maximum or minimum of the underlying achieved through the
period. The payoff from the lookback call is the terminal price of the undelying less the minimum value

For this particular option an analytical solution has been found, due to Goldman et al. (1979), which is shown in formula 13.1 and implemented in code 13.3

Monte Carlo simulation can be used to price a lot of different options. The limitation is that the options should be European. American options can not be priced by simulation methods. In chapter 11 we looked at a general simulation case where we wrote a generic routine which we passed a payoff function to, and the payoff function was all that was necessary to define an option value. The payoff function in that case was a function of the terminal price of the underlying security. The only difference to the previous case is that we now have to generate a price sequence and write the terminal payoff of the derivative in terms of that, instead of just generating the terminal value of the underlying security from the lognormal assumption.

Recall that one will generate lognormally distributed variables as

where the current time is and terminal date is . To simulate a price sequence one splits this period into say periods, each of length

Each step in the simulated price sequence is

Code 13.4 shows how one would simulate a sequence of lognormally distributed variables.

This code is then used in the generic routine to do calculations, as shown in code 13.5.

To price an option we are then only in need of a definition of a payoff function. We consider a couple of examples. One is the case of an Asian option, shown in code 13.6.

Another is the payoff for a lookback, shown in code 13.7

Code 13.8 shows the Black Scholes price used as a control variate. An alternative could have been the analytical lookback price, or the analytical solution for a geometric average price call shown earlier.

Exotic options are covered in Hull (2003). Rubinstein (1993) has an extensive discussion of analytical solutions to various exotic options.

2003-10-22